Solving Systems Of Equations By Substitution Method Step-by-Step Guide

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When faced with a system of equations, the substitution method provides a powerful algebraic technique to find the solutions. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. In this comprehensive guide, we will delve into the intricacies of the substitution method, illustrating its application with a step-by-step example. Our focus will be on solving the system of equations:

2y + 5x = 13
2y - 3x = 5

Our goal is to determine the correct ordered pair (x, y) that satisfies both equations simultaneously. We'll explore each step meticulously, ensuring a clear understanding of the process.

Understanding the Substitution Method

The substitution method revolves around isolating one variable in one equation and then substituting that expression into the other equation. This process transforms the system into a single equation with a single variable, making it solvable. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. This systematic approach ensures we arrive at the correct solution, which represents the point where the lines represented by the equations intersect on a graph.

Step-by-Step Solution

Let's apply the substitution method to solve the given system of equations:

1. Choose an Equation and Isolate a Variable

Examine both equations and identify the one that is easiest to solve for one of the variables. In this case, both equations have a '2y' term, making it convenient to isolate 'y' in either equation. Let's choose the second equation:

2y - 3x = 5

To isolate 'y', we add '3x' to both sides:

2y = 3x + 5

Now, divide both sides by 2:

y = (3/2)x + 5/2

We now have 'y' expressed in terms of 'x'. This is a crucial step in the substitution method, as it allows us to replace 'y' in the other equation with this expression. The key here is to choose the equation that allows for the simplest isolation of a variable, minimizing fractions and complex manipulations.

2. Substitute the Expression into the Other Equation

Next, we substitute the expression we found for 'y' (which is (3/2)x + 5/2) into the other equation (the one we didn't use in step 1). This is the heart of the substitution method, where we reduce the system of two equations into a single equation with one variable. Substituting into the same equation we used to isolate 'y' would be a circular process and wouldn't help us solve the system.

The first equation is:

2y + 5x = 13

Substitute 'y' with '(3/2)x + 5/2':

2((3/2)x + 5/2) + 5x = 13

This substitution is a critical step. It replaces 'y' with its equivalent expression in terms of 'x', effectively reducing the problem to a single equation with only 'x' as the variable. This allows us to solve for 'x' directly.

3. Solve for the Remaining Variable

Now we have an equation with only 'x':

2((3/2)x + 5/2) + 5x = 13

Distribute the 2:

3x + 5 + 5x = 13

Combine like terms:

8x + 5 = 13

Subtract 5 from both sides:

8x = 8

Divide both sides by 8:

x = 1

We have successfully found the value of 'x'! This is a significant milestone in solving the system. The algebraic manipulations, such as distributing, combining like terms, and isolating 'x', are fundamental skills in solving equations.

4. Substitute Back to Find the Other Variable

Now that we know 'x = 1', we can substitute this value back into either of the original equations or the expression we found for 'y' in terms of 'x'. The goal is to find the corresponding value of 'y'. Let's use the expression we found in step 1:

y = (3/2)x + 5/2

Substitute 'x = 1':

y = (3/2)(1) + 5/2
y = 3/2 + 5/2
y = 8/2
y = 4

Thus, we have found that 'y = 4'. Substituting the known value of 'x' back into a previous equation is a common technique in algebra. It allows us to leverage our progress to find the remaining unknowns. The choice of which equation to use for back-substitution is often a matter of convenience, aiming for the equation that minimizes calculations.

5. Write the Solution as an Ordered Pair

The solution to the system of equations is the ordered pair (x, y), which represents the point where the two lines intersect. We found that 'x = 1' and 'y = 4', so the solution is:

(1, 4)

This ordered pair represents the single point that satisfies both equations in the system. Geometrically, it's the intersection point of the two lines represented by the equations. The ordered pair notation is a standard way of expressing solutions to systems of equations, clearly indicating the values of 'x' and 'y'.

Verify the Solution

To ensure our solution is correct, we can substitute the values of 'x' and 'y' back into both of the original equations. If both equations hold true, then our solution is verified.

Equation 1:

2y + 5x = 13

Substitute 'x = 1' and 'y = 4':

2(4) + 5(1) = 13
8 + 5 = 13
13 = 13  (True)

Equation 2:

2y - 3x = 5

Substitute 'x = 1' and 'y = 4':

2(4) - 3(1) = 5
8 - 3 = 5
5 = 5  (True)

Since the ordered pair (1, 4) satisfies both equations, it is indeed the correct solution to the system. Verification is a crucial step in problem-solving, especially in mathematics. It provides confidence in the accuracy of the solution and helps catch any potential errors made during the solving process. By substituting the solution back into the original equations, we ensure that the values of 'x' and 'y' are consistent across the entire system.

Conclusion

Through the detailed step-by-step process, we have successfully solved the system of equations using the substitution method. The correct ordered pair is (1, 4), which corresponds to option B. This method provides a robust approach to solving systems of equations, particularly when one variable can be easily isolated. Mastering the substitution method is crucial for students studying algebra and provides a strong foundation for more advanced mathematical concepts. Remember to always verify your solution to ensure accuracy and a complete understanding of the problem-solving process.

When solving a system of equations, the goal is to find the values for the variables that satisfy all equations simultaneously. These values are often represented as an ordered pair (x, y). Understanding how to identify the correct ordered pair is crucial for success in algebra and beyond. Let's consider the system of equations we solved earlier:

2y + 5x = 13
2y - 3x = 5

We found the solution to be (1, 4). Now, let's explore why this is the correct ordered pair and how it differs from the other options provided:

  • A. (-1, 9)
  • B. (1, 4)
  • C. (-3, 14)
  • D. (3, 7)

Understanding Ordered Pairs

An ordered pair (x, y) represents a point on a coordinate plane. In the context of systems of equations, it represents the point of intersection of the lines represented by the equations. The 'x' value is the horizontal coordinate, and the 'y' value is the vertical coordinate. For an ordered pair to be a solution to a system of equations, it must satisfy all equations in the system. This means that when you substitute the 'x' and 'y' values into each equation, the equation must hold true.

Why (1, 4) is the Correct Solution

We've already demonstrated through the substitution method and verification that (1, 4) is the solution to the system of equations. Let's reiterate why:

  • Equation 1: 2y + 5x = 13

    Substituting x = 1 and y = 4:

    2(4) + 5(1) = 8 + 5 = 13 (True)
    
  • Equation 2: 2y - 3x = 5

    Substituting x = 1 and y = 4:

    2(4) - 3(1) = 8 - 3 = 5 (True)
    

Since (1, 4) satisfies both equations, it is the correct solution. It represents the single point where the lines described by these equations intersect.

Why Other Options are Incorrect

To understand why the other options are incorrect, we need to test them by substituting the 'x' and 'y' values into the equations. If an ordered pair does not satisfy both equations, it is not a solution to the system.

A. (-1, 9)

  • Equation 1: 2y + 5x = 13

    2(9) + 5(-1) = 18 - 5 = 13 (True)
    
  • Equation 2: 2y - 3x = 5

    2(9) - 3(-1) = 18 + 3 = 21 (False)
    

Since (-1, 9) does not satisfy the second equation, it is not a solution.

C. (-3, 14)

  • Equation 1: 2y + 5x = 13

    2(14) + 5(-3) = 28 - 15 = 13 (True)
    
  • Equation 2: 2y - 3x = 5

    2(14) - 3(-3) = 28 + 9 = 37 (False)
    

Since (-3, 14) does not satisfy the second equation, it is not a solution.

D. (3, 7)

  • Equation 1: 2y + 5x = 13

    2(7) + 5(3) = 14 + 15 = 29 (False)
    
  • Equation 2: 2y - 3x = 5

    2(7) - 3(3) = 14 - 9 = 5 (True)
    

Since (3, 7) does not satisfy the first equation, it is not a solution.

Key Takeaway: Ordered Pair for System of Equations

The key takeaway is that an ordered pair is only a solution to a system of equations if it satisfies every equation in the system. If an ordered pair fails to satisfy even one equation, it is not a solution. This principle is fundamental to understanding systems of equations and their graphical representation. The point of intersection of the lines represents the unique ordered pair that satisfies both equations simultaneously.

Methods for Finding the Correct Ordered Pair

Besides the substitution method we used earlier, there are other methods for finding the correct ordered pair:

  1. Substitution Method: This involves solving one equation for one variable and substituting that expression into the other equation. This is the method we demonstrated in detail earlier.

  2. Elimination Method: This involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This method is particularly useful when the coefficients of one variable are opposites or easily made opposites.

  3. Graphing Method: This involves graphing both equations on the same coordinate plane. The point where the lines intersect represents the solution to the system. While this method provides a visual representation of the solution, it may not be as precise as algebraic methods.

  4. Testing Ordered Pairs: As we demonstrated above, you can test ordered pairs by substituting the values into the equations. This is a useful method for verifying a solution or for multiple-choice questions where the possible solutions are provided.

Tips for Choosing the Correct Ordered Pair

Here are some tips to help you choose the correct ordered pair when solving systems of equations:

  • Always verify your solution: Substitute the ordered pair back into the original equations to ensure it satisfies both.
  • Be careful with signs: Pay close attention to positive and negative signs when substituting values.
  • Consider the context: In some real-world problems, the solution must make sense in the given context. For example, a solution with negative values might not be appropriate in a situation involving quantities.
  • Use estimation: If you are graphing the equations, you can use estimation to approximate the point of intersection and narrow down the possible solutions.

Conclusion

In conclusion, understanding how to identify the correct ordered pair is essential for solving systems of equations. The ordered pair must satisfy all equations in the system, representing the point of intersection of the lines. By using methods like substitution, elimination, or graphing, and by carefully verifying your solution, you can confidently find the correct ordered pair. In our example, the correct ordered pair is (1, 4), which satisfies both equations in the system.